A formability index for the deep drawing of stainless steel

Abstract The effects of process parameters on the formability of the deep drawing of rectangular cups made ofSUS304 stainless steel were investigated by both the finiteelement analysis method and the experimental approach. Astatistical analysis was employed to construct an orthogonalchart which reflects the effects of the process parametersand their interactions on the formability of rectangular cupdrawing. The material properties and the forming limitdiagram (FLD) of SUS304 stainless steel were obtainedfrom the experiments conducted in the present study andwere employed by the finite element simulations. In thefinite element analysis, the strain path that led to fracture inthe drawing process was examined and the failure modescaused by different process parameters were also identified.With the help of statistical analysis, a formability index forthe deep drawing of SUS304 stainless steel rectangularcups was constructed and the critical value of theformability index was determined from the finite elementsimulation results. The actual drawing processes of rectangular cups were also performed in the present study. Thevalidity of the finite element simulations and the formability index were confirmed by the good agreement betweenthe simulation results and the experimental data. Theformability index proposed in the present study provides aconvenient design rule for the deep drawing of SUS304stainless steel rectangular cups.Keywords Rectangular cup drawing . Process parameters .Finite element analysis . Statistical analysis . FormabilityindexF.-K. Chen (*) : S.-Y. LinDepartment of Mechanical Engineering,National Taiwan University,Taipei, Taiwan, Republic of Chinae-mail: fkchen@ntu.edu.tw

1 IntroductionRecently, deeply drawn rectangular cups have been widelyapplied to the electronics industry, such as for lithiumbattery cases. Different materials have been selected for theapplications. Among them, stainless steel is most often useddue to its superior corrosion resistance property, though theformability of stainless steel is not so preferable to othermetals. Although the geometry of a rectangular cup issimple, the material flow pattern during the drawingprocess is quite complicated. There are many processparameters that may affect the material flow in therectangular cup drawing, such as material properties andthe geometry of the sheet blank, punch radius and diecorner radius etc. For some deeply drawn cups, more thanone drawing process to produce the parts may be needed.Hence, in the process design of a rectangular cup withgiven dimensions, a formability index which can predictwhether the cup can be successfully formed by one singledrawing process is always desired.A lot of research effort has been made to investigate thedeep drawing of rectangular cups. Kuwabara et al. [1]examined the effects of the cup geometries on theformability of the deep drawing of square cups. Theyanalysed the failure modes and suggested an optimumdesign to prevent to drawn cup from fracture. Mori andMarumo [2] investigated the material flow pattern in thesquare cup drawing process with different lubricationconditions and cup geometries. Danckert [3] performeddifferent experiments to measure the strain distribution in asquare cup drawing process. Since the finite elementmethod was widely applied to the analysis of metal-formingprocesses in 1980s [4], the material flow in the formingprocess could be predicted easily from computer simulations. Chung et al. [5] analysed the effect of planar

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2 Mechanical properties tests

Tensile tests were performed to determine the stressstraincurve and hemi-spherical punch drawing tests were conducted to construct the FLD for SUS304 stainless steel. Inthe tensile tests, the specimens made of 0.61-mm-thickSUS304 sheet were prepared according to the ASTMstandards. The specimens were cut along planes coincidingwith the rolling direction (0) and at angles of 45 and 90to the rolling direction. The cut edges were polished toavoid fracture occurring at an undesired location of thespecimen. The average flow stress A calculated accordingto A A0 2A45 A90 =4 was used to establish thestressstrain relations, where 0, 45 and 90 are the stressesobtained from the specimens cut in the rolling, 45 and 90directions, respectively. Since true stresses and true strainsare used in the finite element simulations, the measuredstresses and strains were converted to true stresses and truestrains, the relations being plotted in Fig. 1.Since Keeler and Backofen [7] introduced the concept ofFLD in 1963, it has been a widely adopted criterion forfracture prediction in sheet-metal forming. To determine anFLD, stretching tests were performed for sheet specimenswith different widths ranging from 20 mm to 140 mm inincrements of 20 mm, using a semi-spherical punch of

1200

true stress (MPa)

anisotropy on the deep drawing of circular and square cups

using the finite element method. Chen and Chuang [6]performed finite element simulations to study the blankdesign for the deep drawing of square cups. However, veryfew literature is found regarding the formability indexmentioned above and, to date, such a formability index hasnot been well defined either theoretically or empirically.In the present study, the three-dimensional finite elementmethod and the experimental analysis were conducted tostudy the effects of process parameters on the formability ofthe deep drawing of rectangular cups made of SUS304stainless steel. In order to facilitate the study, the materialproperties and the forming limit diagram (FLD) of SUS304stainless steel were obtained from experiments and werethen employed by the finite element simulations to studythe material flow and failure modes in a rectangular cupdrawing process. A statistical analysis was also employedto establish the interactive relationships among theseprocess parameters and to construct an empirical formability index for the deep drawing of rectangular cups made ofSUS304 stainless steel. In addition, the actual drawingprocesses of rectangular cups were performed to validatethe finite element simulations and the formability index,and the experimental data were compared with thefinite element simulation results both qualitatively andquantitatively.

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0.1

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true strainFig. 1 Stressstrain relation for SUS 304 stainless steel

78 mm in diameter. The specimens were first electrochemically etched with circular grids that would be deformedinto ellipses after being stretched. The engineering strainsmeasured in the major and minor axes of the ellipse aretermed the major strain and minor strain, respectively.They are also the principal strains on the planes where thestrains are measured. The major and minor strains measuredin the location closest to the fracture for each specimenwere recorded and were then plotted against one anotherwith the major strain as the ordinate. The curve fitted to thestrain points was defined as the forming limit curve, alsotermed the failure curve. Considering the safety factor forthe design purpose, a 10% off-set downward of the failurecurve is adopted as the design curve. For the finite elementsimulation purpose, the engineering major and minorstrains were converted to true major and minor strains,and both the failure curve and the design curve are plottedin Fig. 2. The design curve was used as the failure criterionfor the prediction of the occurrence of fracture in the finiteelement analysis.

3 Finite element analysis

In the present study, the tooling geometries were constructed by the CAD program PRO/Engineer and were thenconverted into the finite element mesh, as shown in Fig. 3,using the program DeltaMESH. The material properties andFLD obtained from the previous tests, a punch velocity of5 m/s and coefficient of Coulomb friction of 0.1 wereadopted in the finite element simulations, the blank-holderforce being varied depending on the blank size. The finiteelement software PAM_STAMP was employed to performthe analysis and the four-node shell element was used in thesimulations.The three-dimensional finite element simulations werefirst performed to examine the effects of process parameterson the formability of the deep drawing of rectangular cups

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Fig. 2 Forming limit diagram

(FLD) for SUS 304 stainlesssteel (t=0.61 mm)

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major strain

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Failure Curve0.6

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Design Curve

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minor strainand the major process parameters that affect the formabilitythe most were identified. The process parameters analysedin the present study include punch radius (Rp), die radius(Rd), die corner radius (Rc), die gap (c) and the length-towidth ratio (a/b), as shown in Fig. 4. One process parameterwas analysed at a time while the other process parametersremained the same. In each simulation, the rectangular cupwas drawn to the presence of fracture according to the FLDconstructed in the previous experiments, and the drawndepth (H) was used as the index for comparison. Inaddition, the low-level and high-level values of eachprocess parameter to be used in the statistical analysis werealso determined from the finite element analysis. Based onboth the statistical analysis and the finite element simulation results, a formability index was proposed for the deepdrawing of rectangular cups made of SUS304 stainlesssteel.The relationships between the punch radius and thedrawn depth obtained from the finite element simulations

are shown in Fig. 5, with the punch radius and drawn depthbeing normalised by the sheet thickness (t). As expected, asmaller punch radius induces an early fracture and results ina smaller drawn depth. The low-level and high-level valuesof 3t and 15t, respectively, were chosen for the statisticalanalysis according to Fig. 5. To examine the failure mode,the major strain and minor strain paths tracing the fracturepoints in the drawing process using punches with radii of2.01 mm and 8.04 mm are shown in Fig. 6a,b, respectively.It is seen in both figures that the fracture of the sheet is dueto biaxial stretching, since both the major and minor strainsare positive. It is also noted in both figures that both the

Fig. 3 Finite element meshes for a rectangular cup drawing process

Fig. 4 Geometric parameters of a rectangular cup drawing process

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Fig. 5 Relation between punch

radius (Rp) and drawn depth (H)

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Fig. 6a, b Major strain and

minor strain paths for the drawings with different punch radii.a Major strain path. b Minorstrain path

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major and minor strains increase more rapidly with a

smaller punch radius than those with a larger punch radius.This implies that the smaller punch profile restrains thematerial from being stretched in both directions, resulting inhigher strains.The effect of die radius on the formability of rectangularcup drawing is shown in Fig. 7. Compared with Fig. 5, theeffect of the die radius is less significant than that of thepunch radius. It is also noted that the drawn depth variesinsignificantly for die radii larger than 7.5t. The low-leveland high-level values for the statistical analysis are chosento be 3t and 12t, respectively.Unlike the monotonically increasing curves shown inFigs. 6 and 7, the relations between the die corner radiusand the drawn depth have a reflection point, as shown inFig. 8, with the die corner radius and drawn depth beingnormalised by the cup width (b). It implies that there existsan optimum die corner radius for the drawing of rectangularcups. The locations of fracture for different die corner radiiare also varied, as indicated in the finite element simulationresults. For the die corner radius of 0.5 mm, the fracturepoint is located at the punch corner, as shown in Fig. 9a,indicating a stretch failure mode, while the fracture pointmoves to the side of the punch, as shown in Fig. 9b, for the

die corner radius of 8 mm, representing a change from the

stretch failure mode to the plane strain failure mode. Sincethe relationships between the die corner radius and thedrawn depth are not represented by a monotonicallyincreasing curve, the statistical analysis should be performed for the increasing region and the decreasing regionseparately. According to Fig. 8, the curve can be separatedinto two regions of 0.05 to 0.18 and 0.18 to 0.32 in the Rc/baxis. However, only the region of 0.05 to 0.18 is discussedin this paper, since this region is more favourable in theprocess design, though the method of analysis is similar forthe other region.The effect of the die gap on the formability of the deepdrawing of rectangular cups is not significant, as shown inFig. 10. The strain paths of the fracture point are alsoalmost identical in the drawing processes with various die32

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Fig. 8 Relation between die corner radius (Rc) and drawn depth (H)

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Fig. 11 Relation between

length-to-width ratio (a/b) anddrawn depth (H)

H/b (mm/mm)

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gaps according to the finite element simulation results. The

effect of the length-to-width ratio (a/b) on the formability isalso insignificant, as shown in Fig. 11. Similar strain pathsof the fracture point were also observed in the drawingprocesses with different values of a/b.According to the finite element analysis, the punchradius and the die corner radius are the major processparameters in a rectangular cup drawing process. The lowlevel and high-level values for each process parametersuggested by the finite element simulation results will beused in the following statistical analysis.

of each process parameter and a combination of them based

on the finite element simulation results. The coefficient ofeach term in the linear equation represents the weight ofthat term affecting the result. The larger the coefficient, themore significant the effect that term has on the result. Thelinear equation generated by the software is given by:

4 Statistical analysis

The coefficients of the omitted terms in Eq. 1 are all much

smaller than 0.1. Hence, it can be determined from Eq. 1 thatthe most important process parameters are B(=Rc/b) and C(=Rp/t), and the interactive effects of the combination ofprocess parameters on the formability of rectangular cupdrawing can be ignored. The parameters Rc/b and Rp/t as

The process parameters considered in the statistical analysis

are normalised by suitable dimensions and are denoted by:A=a/b, B=Rc/b, C=Rp/t, D=Rd/t and E=c/t. The low-leveland high-level values of each process parameter chosen forthe statistical analysis are summarised in Table 1, assuggested by the finite element analysis. In order tounderstand how the process parameters and their interactions affect the formability of the deep drawing ofrectangular cups, an orthogonal chart based on the lowlevel and high-level values was established as listed inTable 2 to perform the finite element simulations in asystematic way. According to Table 2, there are 16simulations to be performed. In each simulation, the lowlevel and high-level values of each process parameter listedin Table 1 were used as the input data, and the maximumdrawn depth (H) without the occurrence of fracture wasrecorded. The statistical analysis software Design-Ease2.0.11 was employed to generate a linear equation in termsTable 1 Low-level and high-level values of the process parameters

Table 2 Orthogonal array of five factors

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Fig. 12 Values of K versus die

corner radius (Rc)

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Rc/b

to be =2, =1 and =0.25, and the proposed formability index has the form:stH2K43Rp Rc b

well as H/b are then selected to construct the formability

index. Since the software can only provide a linear equationto illustrate the weight of effect of the process parameters, itis not sufficient for constructing the formability index, whichmay be a nonlinear function. An iterative approach was thenadopted to develop the formability index. To start theiteration, a power form consisting of the selected processparameters was proposed for the formability index (K), asgiven by:

RpHRcK2btb

Figure 12 shows the values of K calculated from Eq. 3

with Rc/b as the abscissa, using the finite elementsimulation results as the base data. It is seen in Fig. 12that the values of K are in the range from 0.9 to 2.5, exceptthose with Rc/b<0.1. The values of K calculated from Eq. 3with Rp/t as the abscissa are also plotted and shown inFig. 13. It is noted that most values are in the range from1.0 to 2.2, except those with Rc/b<0.1, which give muchhigher values. It suggests that the critical value of K for thedeep drawing of SUS304 stainless steel rectangular cupsmay be chosen as 2.2 for Rc/b>0.1. That is, for a SUS304rectangular cup, if its dimensions result in a value of Kbeing larger than 2.2 according to Eq. 3, it may not besuccessfully formed by one drawing process. The proposedformability index will be validated by the experimentsdescribed below.

The results of the finite element simulations performed

for the statistical analysis were used as the base data todetermine the values of , and . Based on the weights ofthe process parameters in affecting the formability, initialvalues of , and were tested to fit the above-mentioneddata according to Eq. 2. The errors were then estimated anda new set of values was assigned to , and for asubsequent trial. This trial-and-error method was repeateduntil an optimum set of values for , and wasdetermined. In the present study, these values were foundFig. 13 Values of K versuspunch radius (Rp)

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K from Rp/bRc/b <0.1

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R p /t

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Fig. 14 Punches used in the drawing tests

5 Experimental validationsIn order to validate the finite element analysis and theproposed formability index, six sets of tooling were manufactured to conduct the rectangular cup drawing tests. Thedimensions of the tooling geometries were selected to be thesame as those adopted in the finite element simulations andwithin the range between the low-level and high-level valuessets for the statistical analysis. The punches used in thedrawing tests are shown in Fig. 14. The SUS304 stainlesssteel sheets, of 0.61 mm and 0.72 mm in thickness, were cutto the designed blank sizes and lubricated with oil for the

Fig. 15a, b Comparison

drawing tests. The experimental results of the drawn-cup

shape, locations of fracture, major and minor strains,thickness distribution and the K values were compared withthose obtained from the finite element simulations for all sixsets of tooling. Since the comparison of the finite elementresults and the experimental data shows a consistent trendfor all six sets of tooling, only the test results of the selectedtooling are demonstrated in this paper.The drawn-cup shapes and their projections of arectangular cup obtained from the experiments and thefinite element simulations are shown in Fig. 15a,b,respectively. It is seen in Fig. 15a that both drawn-cupshapes are in a good agreement with each other and so arethe projections shown in Fig. 15b. In addition, both theexperimental and the finite element simulation resultsindicate that the fracture occurs at the punch corner for asmaller punch radius. The major and minor strains of thedrawn rectangular cup are compared in Fig. 16a,b,respectively. Both strains are measured along the diagonalfrom the centre of the drawn cup. As seen in both figures,the finite element simulation results agree well with theexperimental data both in trend and in magnitude, thoughan insignificant difference in magnitude is noted. In order tomeasure the thickness of a drawn cup, a 3-mm-wide stripwas wire-cut along the diagonal of the cup and thethickness was measured with a micrometer. The measuredthickness distribution of a square cup (a/b=1) was com-

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Fig. 16a, b Major and minor strains measured from experiments and simulations. a Major strain. b Minor strain

pared with that obtained from the finite element simulation,

as shown in Fig. 17. Both the experimental data and thefinite element simulation results indicate that the thinnestportion is at the punch corner and the sheet at the flange isthickened. The thickness distribution predicted by the finite

element simulation is found consistent with that obtained in

the experiment, as shown in Fig. 17.The above comparisons have demonstrated the accuracyof the finite element analysis performed for the deepdrawing processes of rectangular cups. The experimental

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distance from square cup center point (mm)

Fig. 17 Thickness distribution along the diagonal of the drawn cup

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demonstrated the usefulness of the formability index

proposed in the present study. The major process parameters affecting the formability of the deep drawing ofrectangular cups are identified to be the punch radius andthe die corner radius, according to the finite elementsimulations results and the statistical analysis. The statistical analysis also indicates that the interaction of the processparameters has an insignificant effect on the formability ofthe deep drawing of rectangular cups.The finite element analysis reveals that there exists anoptimum die corner radius for the rectangular cup drawingprocess. The formability is improved with the increase ofdie corner radius up to an optimum value and then becomesworse when the die corner radius increases further. Thefailure mode analysis shows that the fracture point movesfrom the punch corner to the side of punch for a die cornerradius larger than the optimum value, representing a changefrom the stretch failure mode to the plane strain failuremode, which is more prone to fracture. Hence, a favourableprocess design for the rectangular cup drawing shouldavoid a die corner radius larger than the optimum value.

data were also employed to validate the proposed formability index. The maximum drawn depth, the sheetthickness and the associated tooling dimensions of eachdrawing process were substituted into Eq. 3 to calculate theK value. The experimental K values are plotted in Fig. 18with Rp/t as the abscissa. It is seen in Fig. 18 that all of theK values are within the range from 1.1 to 2.1, which areclose to those predicted by the finite element simulationsbut with a smaller maximum value of K. It indicates that theformability index given by Eq. 3 is a valid form and thecritical K value can be set conservatively as 2.0 for the deepdrawing of SUS304 stainless steel rectangular cups. Theexperimental results confirm the validity of both the finiteelement simulations and the statistical analysis.

6 Concluding remarksThe experimental results have validated the finite elementsimulations performed for the deep drawing of rectangularcups made of SUS304 stainless steel and have also

The formability index (K) proposed in the present study

provides a convenient design rule for the deep drawing ofSUS304 stainless steel rectangular cups. If the value of Kcalculated from Eq. 3 with the given rectangular cupdimensions is larger than a critical value, fracture is likelyto occur before the cup is completely drawn, and multipledrawings may be required to produce a sound rectangularcup. The critical K value for SUS304 stainless steel has beendetermined in the present study by both the finite elementanalysis and the experimental approach. It should be notedthat both the form of the proposed formability index and thecritical value of K may be varied for other materials, sincethe material properties are not considered in the proposedform. However, the method of approaches to construct theformability index can be applied to other drawing processeswith different materials. The formability index proposed inthe present study provides a convenient design rule for thedeep drawing of SUS304 stainless steel rectangular cups.Acknowledgements The authors would like to thank the NationalScience Council of the Republic of China for financially supportingthis research under contract no. NSC 89-2212-E-002-147, which

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makes the experimental work possible. They are also grateful to ESI inrunning the PAM_STAMP program.